Optimal Reinsurance-Investment Problem under a CEV Model: Stochastic Differential Game Formulation

This paper focuses on a stochastic diﬀerential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big company has suﬃcient asset to invest in the risky asset which is described by the constant elasticity of variance (CEV) model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. In this paper, we describe the Nash equilibrium of the game and prove a veriﬁcation theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Furthermore, numerical examples are presented to show our results.


Introduction
Recently, most insurance companies manage their business by means of reinsurance and investment, which are effective way to spread risk and make profit. erefore, these have inspired hundred researches. For instance, Schmidli [1], Promislow and Young [2], and Bai and Guo [3] investigated the optimal problems for an insurance company in the case of minimizing the ruin probability. Yang and Zhang [4], Wang [5], and Cao and Wan [6] studied the optimal reinsurance and investment problems of expected utility maximization. e latest researches on insurance and investment management problem can be referred to Yu et al. [7], Zhang et al. [8], Peng et al. [9], Yu et al. [10], Ruan et al. [11], Yu et al. [12], Huang et al. [13], Zeng et al. [14], Li et al. [15] and references therein.
However, most of the literature mentioned above only considered one insurance company, while there are many insurance companies in the market in reality and they compete with each other. us, two insurance companies, a big one and a small one, are focused on in this paper. is competition between the two insurance companies can be formulated as a stochastic differential game. In previous researches, Suijs et al. [16] showed that problems in non-life insurance and non-life reinsurance can be modeled as cooperative games. Zeng [17] discussed the competition between two companies and contrasted a single payoff function which depended on both insurance companies' surplus processes. Taksar and Zeng [18] investigated stochastic differential games between two insurance companies who employed the reinsurance to reduce risk exposure. Some papers focus on the relative performance of two insurance companies under a nonzero sum stochastic differential game framework, such as Bensoussan et al. [19], Meng et al. [20], Pun and Wong [21], and Siu et al. [22]. However, some literatures related to stochastic differential games ignore the problem of investment in insurance companies. Nowadays, investment plays a significant role in the insurance business, especially for those big insurance companies who have enough ability to invest in the risky asset for more profits.
Another aspect worthy to be further explored is that the price processes of risky assets in most of literature about the optimal reinsurance and investment problems in frameworks of stochastic differential games are assumed to follow a geometric Brownian motion (GBM), which implies that the volatilities of risky assets are constant and deterministic.
is is contrary to practice according to the empirical results. erefore, many works proposed various stochastic volatility models, such as constant elasticity of variance (CEV) model (Cox and Ross [23]), Stein-Stein model (Stein and Stein [24]), Heston model (Heston [25]) and so on. Among these stochastic volatility models, the CEV model is a natural extension of the GBM model and has the ability of capturing the implied volatility skew and explaining the volatility smile.
ere is a great deal of literature documenting the CEV model in assets' return for the optimal investment problem. For example, Gu et al. [26] considered the proportional reinsurance and investment problem for the diffusion risk model under the CEV model. Liang et al. [27] and Lin and Li [28] used the CEV model to study the proportional reinsurance and investment problem for an insurance company with the jump-diffusion risk model. Gu et al. [29] derived the excess-of-loss reinsurance and investment strategies with the risky asset's price following the CEV model. Zheng et al. [30] considered the robust optimal portfolio and proportional reinsurance for an insurer under a CEV model.
As far as we know, there is few research investigating more than one insurance company under the CEV model. erefore, in this paper, we consider a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim processes is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. e big insurance company has more initial surplus than the small one, so the big company has sufficient asset to invest in the risky asset which is described by the CEV model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can only invest in the risk-free asset and purchase reinsurance. e game studied here is zero-sum where there is a single exponential utility. e big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. Firstly, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Finally, numerical simulation are proposed to illustrate the impacts of the model parameters on the strategies. rough this paper, we find that (1) for such a game the small insurance company takes extreme or trivial strategy, i.e., the optimal reinsurance strategy of the small company is either 1 or 0; (2) the optimal reinsurance and investment strategies of the big company are independent of the wealth process X u 1 ,u 2 (t); (3) the effects of some model parameters on the big company's optimal reinsurance strategy are related to the correlation coefficient between the big and small companies' risk processes.
is paper proceeds as follows. In Section 2, we introduce the formulation of our model, describe the Nash equilibrium of the game, and prove a verification theorem for the exponential utility under the CEV model. Section 3 provides the optimal reinsurance and investment strategies of the big insurance company and the optimal reinsurance strategy of the small one in the stochastic differential game. In Section 4, numerical simulations are presented to illustrate our results. Section 5 concludes this paper.

Model Formulation
In this paper, we model the surplus process of the insurance company as where a, b are positive constants and W(t) is a standard Brownian motion. To reduce the risk exposure, the insurance company is allowed to purchase the reinsurance and p(t) represents the proportion of each claim paid by the insurance company. Assume λ(1 − p(t)) is the rate at which the premiums are diverted to the reinsurance company. As usual, we have λ ≥ a. Otherwise the insurance company will make a full reinsurance to receive a positive return with any risk. Considering the reinsurance, the surplus process of the insurance company becomes In this paper, we consider a stochastic differential game played between two insurance companies, a big one and a small one. e big insurance company has more initial surplus than the small one, so the big company has sufficient asset to invest in the risky asset and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can only invest in the risk-free asset and purchase reinsurance, i.e., the reinsurance strategies of the big and small companies p 1 (t) and p 2 (t) satisfy 0 ≤ p 1 (t), 0 ≤ p 2 (t) ≤ 1. e game considered here is zero-sum where there is a single exponential utility. e big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. One company's decision is assumed to be completely observed by its opponent.
Let (Ω, F, (F t ), P) be a complete probability space with filtration (F t ) and two standard Brownian motions W 1 (t) and W 2 (t), adapted to (F t ) with F � F T , where T is a fixed and finite time horizon. e surplus processes of the two insurance companies associated with the proportional reinsurance p i (t), i � 1, 2 are given by Mathematical Problems in Engineering In this paper, the big insurance company is allowed to invest in a risk-free asset whose price process satisfies and a risky asset whose price process is described by the CEV model (cf. Cox [31]): S(0) � s 0 , where r 0 is the interest rate and μ, σS β (t), and β are the appreciation rate, the instantaneous volatility, and the elasticity parameter of the risky asset. W 3 (t) is a standard Brownian motion independent of W 1 (t) and W 2 (t). As usual, we assume that μ > r 0 and β satisfies the general condition β ≥ 0. Meanwhile, the small one can invest in a risk-free asset to avoid risk. Strategies u 1 ≔ (π(t), p 1 (t)) and u 2 ≔ p 2 (t) are said to be admissible if they are (F t )− progressively measurable and satisfy (u 1 , u 2 ) ≔ ((π(t), p 1 (t)), p 2 (t)) ∈ Π: where π(t) represents the amount invested in the risky asset by the big insurance company at time t. Here Π is called the admissible set. Corresponding to an admissible strategy and the initial wealth x 1 > x 2 , the wealth processes of the big and small insurance companies X u 1 1 (t) and X u 1 ; then X u 1 ,u 2 (t) follows the following stochastic differential equation: Here, we usually assume x > 0, which describes that one of the two insurance companies is big, and the other is small. at is, this model is suitable for the case that x is positive and we can distinguish the big and small companies easily. If x is very small, we will choose the insurance company with larger initial wealth as the big one, and the other is the small one. If it is difficult to distinguish the big company and small company from the initial wealth, this framework may not be suitable.

Remark 1.
If the elasticity parameter β � 0 in equation (1), the CEV model reduces to the GBM model. In this paper, we consider the exponential utility which is given by where c > 0. e exponential utility has the constant absolute risk aversion parameter c, which plays an important role in insurance mathematics and actuarial practice.
the strategy (u * 1 , u * 2 ) is said to achieve a Nash equilibrium or equivalently a saddle point for the game, if the following inequalities are satisfied. For all (u 1 , u 2 ) ∈ Π, In addition, let denote the lower and upper values of the game respectively. If J(t, s, x) � J(t, s, x), the value function of the game is given by In this paper, the goal of the big insurance company is to maximize the above expected exponential utility function while simultaneously the small one wants to minimize it. For Mathematical Problems in Engineering convenience, we first provide some notations.
and define a variation operator: for any ϕ(t, s, For any given strategy u 2 by the small insurance company, let V u 2 (t, s, x) be the optimal expected exponential utility function of the big insurance company, i.e., en V u 2 (t, s, x) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation Similarly, let V u 1 (t, s, x) be the optimal expected exponential utility function of the small insurance company with any strategy u 1 given by the big insurance company, then V u 1 (t, s, x) satisfied another HJB equation Denote Assume that a saddle point exits, then the game have a value function be the solution to the following equations: and substitute them into equations (16) and (17), respectively, we obtain the following equations: inf with the boundary condition V(T, s, x) � U(x).

Theorem 1 (Verification theorem).
If there exists a con- satisfy equations (20) and (21) with the following moment properties In addition, the parameters satisfy one of the following conditions: is the optimal strategy and the optimal value function is e above theorem guarantees the solution to equations (20) and (21) is the value function for the game.

Solution to the Model
In this section, we solve the game under the expected exponential utility. e big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while the small company is trying to minimize the same quantity to reduce its disadvantage, i.e., 4 Mathematical Problems in Engineering Assume the Nash equilibrium exits and let H(t, s, x) satisfy then H(t, s, x) is the solution to equations (20) and (21) with the boundary condition H(T, s, x) � (x) and H(t, s, x) is the optimal value function according to eorem 1. e following theorem gives the optimal reinsurance and investment strategies of the big company and the reinsurance strategy of the small company in the game under the CEV model with the expected exponential utility.

Theorem 2.
For the problem of maximizing the expected exponential utility for the big company while minimizing it for the small company under the CEV model, the optimal reinsurance and investment strategies are given as follows: )) ≥ − (c/2) and ρ 12 ≥ 0, the optimal reinsurance and investment strategies of the big company and the reinsurance strategy of the small company are e value function is given by equation (B.21).
, the optimal reinsurance strategies of the big and small companies are while the optimal investment strategy of the big company is the same as that in equation (11) and the value function is given in equation (B.25).
the optimal reinsurance strategies of the big and small companies are while the optimal investment strategy of the big company is the same as that in equation (11) and the value function is given in equation (B.28).
, the optimal reinsurance strategies of the big and small companies are e optimal investment strategy of the big company is expressed as that in equation (11) and the value function is given by equation (B.32).
))) and ρ 12 ≥ 0, the optimal reinsurance strategies of the big and small companies are while the optimal investment strategy of the big company is the same as that in equation (11) and the value function is given in equation (B.36).
)), ρ 12 < 0 and − (λ 1 /cb 1 b 2 ρ 12 ) ≤ 1, the optimal reinsurance strategies of the big and small companies are e optimal investment strategy of the big company is expressed as that in equation (11) and the value function is given by equation (B.39).
)), ρ 12 < 0, − (λ 1 /cb 1 b 2 ρ 12 ) > 1, and e r 0 T < − (λ 1 /cb 1 b 2 ρ 12 ), the optimal reinsurance strategies of the big and small companies are the same as those in equation (29), while the optimal investment strategy of the big Mathematical Problems in Engineering company is the same as that in equation (27) and the value function is given in equation (B.39).

Proof. See Appendix B.
□ Remark 2. From eorem 2 we conclude that the wealth has no influence on the optimal investment strategy. is can be explained by the risk tolerance of the exponential utility function. e risk tolerance is − U x /U xx � 1/c, which is independent of the wealth. us, the optimal strategy is independent of the wealth. In addition, the optimal reinsurance strategy of the big company is not related to the wealth as well and that of the small company is either 1 or 0, i.e., the small insurance company takes extreme or trivial strategy in this game.

Remark 3.
In the case that the risky asset's price follows the GBM model, the optimal strategy of the big insurance company is Compared with equation (34), we see that the optimal investment strategy under the CEV model can be decomposed into two parts. One is which is similar to the optimal strategy under the GBM model, but the volatility is stochastic. us, we call M(t) as the moving GBM strategy. e other one is which reflects the insurance company's decision to hedge the volatility risk and we regard it as a correction factor. e following corollary discusses the property of the correction factor.

Corollary 1. e correction factor N(t) is a monotone decreasing function with respect to time t and satisfies
Proof. According to μ > r 0 and β > 0, we derive N t � − β(μ − r 0 )e − 2r 0 β(T− t) < 0. It implies that the correction factor is a monotone decreasing function with respect to time t. Since and N(T) � 1, we obtain inequality (37).
□ Corollary 1 shows that the correction factor advises the big insurance company to invest more wealth in the risky asset at the beginning of the investment horizon and steadily decrease the amounts as time goes on.

Remark 4.
We find that the optimal investment strategy is independent of the optimal reinsurance strategies. e main reason is that in this model, we assume that the financial market is not affected by the insurance market, which is used in a great deal of existing literature, such as Bai and Guo [3], Gu et al. [26], Gu et al. [29], and so on.

Numerical Simulations of the Big Company's Optimal
Reinsurance Strategy. Figure 1 illustrates the influence of the risk averse coefficient c on the optimal reinsurance strategy p * 1 (t). e relationship between p * 1 (t) and c is negative. is can be attributed to the fact that a larger c means the big insurance company is a more risk-averse individual. With the increase of c, the big company wants to purchase more reinsurance to avoid risk and undertake less risk itself. Figure 2 shows the effect of r 0 on the optimal reinsurance strategy p * 1 (t). As r 0 increases, the big insurance company will obtain more profit from investment in the risk-free asset. erefore, it has more money to purchase the reinsurance and bear less risk itself, so p * 1 (t) decreases with r 0 . Figure 3 indicates the impact of λ 1 on the optimal reinsurance strategy p * 1 (t). We can see that a greater λ 1 yields a greater reinsurance strategy. is is because that as λ 1 increases, the cost of reinsurance will become more expensive and the big insurance company prefers to maintain a stable revenue by purchasing less reinsurance and undertake more by itself.
As is shown in Figure 4, we find that p * 1 (t) is a decreasing function of b 1 . is can be explained by that b 1 implies the fluctuation of the big insurance company's claim process. When b 1 increases, the insurance company wants to purchase more reinsurance while undertaking less risk by itself.
From Figure 5, we can see if ρ 12 > 0, b 2 exerts a positive effect on the optimal reinsurance strategy p * 1 (t) and if ρ 12 < 0, the effect is opposite. is can be attributed to that the fluctuations of the big and small companies' claim processes are more serious with b 2 rising when ρ 12 > 0. erefore, the big company will take more risk, while when ρ 12 < 0, the fluctuation of the big insurance company's claim process is weaker with the volatility of the small company becoming stronger. So the big company will face less risk.
In Figure 6, we demonstrate the effect of the correlation coefficient between the big and small insurance companies' risk processes ρ 12 on the optimal reinsurance strategy p * 1 (t). We find that no matter ρ 12 is positive or negative, the higher ρ 12 is, the bigger p * 1 (t) is. is is because that for the big company, the more relevant the relationship between two companies is, the greater the influence of the small company on the big company is. So the big company has to undertake more risk with ρ 12 rising. Figure 7(a) plots the evolution of the risky asset's price over time under the CEV model. According to the change trend of the risky asset's price, we plot the dynamic behaviors of the optimal investment strategy of the big insurance company in Figure 7(b) and we can see that the change trend of the   Figure 4: (a) e effect of b 1 on the big company's optimal reinsurance strategy p * 1 (t) when ρ 12 > 0. (b) e effect of b 1 on the big company's optimal reinsurance strategy p * 1 (t) when ρ 12 < 0.

Mathematical Problems in Engineering
optimal investment strategy is opposite to that of the risky asset's price, which can be attributed to the expression of the CEV model. When the price of risky asset is high, the big company should be cautious to invest, for the reason that the higher the risky asset's price is, the higher risk they will undertake.
In Figure 8(a), we find that the rate of the risky asset's return μ exerts positive effect on the optimal investment strategy π * (t). is is consistent with intuition. As μ increases, the big company will obtain more from investment. erefore, it will increase the amounts invested in the risky asset. As shown in Figure 8(b), the optimal investment strategy π * (t) is a decreasing function of the interest rate r 0 . When the interest rate r 0 increases, the risk-free asset is more attractive. en the big company will invest more in the risk-free asset and reduce investment in the risky asset. Figure 9(a) indicates the impact of the risk averse coefficient c on the optimal investment strategy π * (t). We see that c exerts negative effect on π * (t), which means that the big company with the higher risk averse level will invest less in the risky asset to avoid risk. Figure 9(b) illustrates the effect of the elasticity coefficient β on the optimal investment strategy π * (t). ere is a negative relationship between π * (t) and β. is can be attributed to that a higher β leads to a larger expected drop in volatility and increased probability of a large adverse movement in the risky asset'a price. us, the big company will invest less in the risky asset as β increases to reduce the risk from the investment in the risky asset.

Conclusion
is paper considers a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. e big insurance company has more initial surplus than the small one, so the big company has sufficient asset to invest in the risky asset which is described by the CEV model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can only invest in the risk-free asset and purchase reinsurance. e game studied here is zero-sum where there is a single exponential utility. e big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. Firstly, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Finally, numerical simulation is proposed to illustrate the impacts of the model parameters on the strategies. rough this paper, we find that (1) for such a game, the small insurance company takes extreme or trivial strategy, i.e., the optimal reinsurance strategy of the small company is either 1 or 0; (2) the optimal reinsurance and investment strategies of the big company are independent of the wealth process X u 1 ,u 2 (t); (3) the effects of some model parameters on the big company's optimal reinsurance strategy is related to the correlation coefficient between the big and small companies' risk processes. In future work, we will consider more complex models, such as both two insurers can invest in the risky asset, or n insurers will participate in the game.

Appendix
A. Proof of eorem 1 Proof. By Itô's formula, we have  Figure 9: (a) e effect of c on the optimal investment strategy of the big insurance company π * (t). (b) e effect of β on the optimal investment strategy of the big insurance company π * (t).
Since the last five terms are square-integrable martingales with zero expectation, taking conditional expectation given (t, s, x) on both sides of the above formula and taking equation (21) into account results that By virtue of Lemma B.1, H u 1 ,u * 2 (τ i ∧ T, S(τ i ∧ T), X u 1 ,u * 2 (τ i ∧ T)), i � 1, 2, .. are uniformly integrable. It is easy to see that the equality holds for u 1 � u * 1 . us, we have s, x). Similarly, we can prove Hence, (u * 1 , u * 2 ) is a saddle point for the game by Definition 1 and according to we can derive (A.8) When u 1 � u * 1 , the inequality in the above formula becomes an equality, i.e., H(t, s, x) is the optimal value function. □ B. Proof of eorem 2 Proof. According to the exponential utility described by equation (9), we try to find the optimal value function in the following way: From equation (B.2), we obtain H xx < 0, the infumum in equation (21) is reached at p * If ((λ 2 b 1 − ρ 12 λ 1 b 2 )/(b 1 b 2 2 (1 − ρ 2 12 ))) ≥ − (c/2), then p 2 (t) < (1/2). We have the optimal reinsurance strategies of the small and big company p * 2 (t) � 1 and Equation (B.7) shows that p 1 (t) ∈ (0, ∞) is equivalent to (1) If ρ 12 ≥ 0, we derive the optimal reinsurance strategies of the big and small companies p * 1 (t), p * 2 (t) �